1.INTRODUCTION 1
1.1The nature of turbulence 1
Irregularity 1
Diffusivity 2
Large Reynolds numbers 2
Three-dimensional vorticity fluctuations 2
Dissipation 3
Continuum 3
Turbulent flows are flows 3
1.2Methods of analysis 4
Dimensional analysis 5
Asymptotic invariance 5
Local invariance 6
1.3The origin of turbulence 7
1.4Diffusivity of turbulence 8
Diffusion in a problem with an imposed length scale 8
Eddy diffusivity 10
Diffusion in a problem with an imposed time scale 11
1.5Length scales in turbulent flows 14
Laminar boundary layers 14
Diffusive and convective length scales 15
Turbulent boundary layers 16
Laminar and turbulent friction 17
Small scales in turbulence 19
An inviscid estimate for the dissipation rate 20
Scale relations 21
Molecular and turbulent scales 23
1.6Outline of the material 24
2.TURBULENT TRANSPORT OF MOMENTUM AND HEAT 27
2.1The Reynolds equations 27
The Reynolds decomposition 28
Correlated variables 29
Equations for the mean flow 30
The Reynolds stress 32
Turbulent transport of heat 33
2.2Elements of the kinetic theory of gases 34
Pure shear flow 34
Molecular collisions 35
Characteristic times and lengths 38
The correlation between v1 and v2 38
Thermal diffusivity 39
2.3Estimates of the Reynolds stress 40
Reynolds stress and vortex stretching 40
The mixing-length model 42
The length-scale problem 44
A neglected transport term 45
The mixing length as an integral scale 45
The gradient-transport fallacy 47
Further esti-mates 49
Recapitulation 49
2.4Turbulent heat transfer 50
Reynolds’ analogy 51
The mixing-length model 51
2.5Turbulent shear flow near a rigid wall 52
A flow with constant stress 54
Nonzero mass transfer 55
The mixing-length approach 55
The limitations of mixing-length theory 57
3.THE DYNAMICS OF TURBULENCE 59
3.1Kinetic energy of the mean flow 59
Pure shear flow 60
The effects of viscosity 62
3.2Kinetic energy of the turbulence 63
Production equals dissipation 64
Taylor microscale 65
Scale relations 67
Spectral energy transfer 68
Further estimates 69
Wind-tunnel turbu-lence 70
Pure shear flow 74
3.3Vorticity dynamics 75
Vorticity vector and rotation tensor 76
Vortex terms in the equations of motion 76
Reynolds stress and vorticity 78
The vorticity equation 81
Vorticity in turbulent flows 84
Two-dimensional mean flow 85
The dynamics of ΩiΩi 86
The equation for ωiωi 86
Turbulence is rota-tional 87
An approximate vorticity budget 88
Multiple length scales 92
Stretching of magnetic field lines 93
3.4The dynamics of temperature fluctuations 95
Microscales in the temperature field 95
Buoyant convection 97
Richardson numbers 98
Buoyancy time scale 99
Monin-Oboukhov length 100
Convec-tion in the atmospheric boundary layer 100
4.BOUNDARY-FREE SHEAR FLOWS 104
4.1Almost parallel，two-dimensional flows 104
Plane flows 104
The cross-stream momentum equation 106
The streamwise momentum equation 108
Turbulent wakes 109
Turbulent jets and mixing layers 110
The momentum integral 111
Momentum thickness 112
4.2Turbulent wakes 113
Self-preservation 113
The mean-velocity profile 115
Axisymmetric wakes 118
Scale relations 119
The turbulent energy budget 120
4.3The wake of a self-propelled body 124
Plane wakes 125
Axisymmetric wakes 127
4.4Turbulent jets and mixing layers 127
Mixing layers 128
Plane jets 129
The energy budget in a plane jet 131
4.5Comparative structure of wakes，jets，and mixing layers 133
4.6Thermal plumes 135
Two-dimensional plumes 136
Self-preservation 141
The heat-flux inte-gral 142
Further results 142
5.WALL-BOUNDED SHEAR FLOWS 146
5.1The problem of multiple scales 146
Inertial sublayer 147
Velocity-defect law 147
5.2Turbulent flows in pipes and channels 149
Channel flow 149
The surface layer on a smooth wall 152
The core region 153
Inertial sublayer 153
Logarithmic friction law 156
Turbulent pipe flow 156
Experimental data on pipe flow 157
The viscous sub-layer 158
Experimental data on the law of the wall 161
Experimental data on the velocity-defect law 162
The flow of energy 163
Flow over rough surfaces 164
5.3Planetary boundary layers 166
The geostrophic wind 166
The Ekman layer 167
The velocity-defect law 167
The surface layer 168
The logarithmic wind profile 169
Ekman layers in the ocean 170
5.4The effects of a pressure gradient on the flow in surface layers 171
A second-order correction to pipe flow 174
The slope of the logarithmic velocity profile 175
5.5The downstream development of turbulent boundary layers 177
The potential flow 179
The pressure inside the boundary layer 181
The boundary-layer equation 182
Equilibrium flow 184
The flow in the wall layer 185
The law of the wall 185
The logarithmic friction law 186
The pressure-gradient parameter 186
Free-stream velocity distributions 188
Boundary layers in zero pressure gradient 190
Transport of scalar contam-inants 194
6.THE STATISTICAL DESCRIPTION OF TURBULENCE 197
6.1The probability density 197
6.2Fourier transforms and characteristic functions 201
The effects of spikes and discontinuities 203
Parseval’s relation 205
6.3Joint statistics and statistical independence 207
6.4Correlation functions and spectra 210
The convergence of averages 211
Ergodicity 212
The Fourier transform of ρ（τ） 214
6.5The central limit theorem 216
The statistics of integrals 218
A generalization of the theorem 220
More statistics of integrals 220
7.TURBULENT TRANSPORT 223
7.1Transport in stationary，homogeneous turbulence 223
Stationarity 223
Staticnary，homogeneous turbulence without mean veloc- ity 224
The probability density of the Lagrangian velocity 226
The Lagrangianintegral scale 229
The diffusion equation 230
7.2Transport in shear flows 230
Uniform shear flow 230
Joint statistics 232
Longitudinal dispersion in channel flow 233
Bulk velocity measurements in pipes 235
7.3Dispersion of contaminants 235
The concentration distribution 235
The effects of molecular transport 237
The effect of pure，steady strain 238
Transport at large scales 241
7.4Turbulent transport in evolving flows 241
Thermal wake in grid turbulence 242
Self-preservation 243
Dispersion rela-tive to the decaying turbulence 245
The Gaussian distribution 246
Disper-sion in shear flows 246
8.SPECTRAL DYNAMICS 248
8.1One- and three-dimensional spectra 248
Aliasing in one-dimensional spectra 248
The three-dimensional spec-trum 250
The correlation tensor and its Fourier transform 250
Two common one-dimensional spectra 251
Isotropic relations 253
Spectra of isotropic simple waves 254
8.2The energy cascade 256
Spectral energy transfer 258
A simple eddy 258
The energy cascade 260
8.3The spectrum of turbulence 262
The spectrum in the equilibrium range 262
The large-scale spectrum 264
The inertial subrange 264
8.4The effects of production and dissipation 267
The effect of dissipation 269
The effect of production 271
Approximate spectra for large Reynolds numbers 272
8.5Time spectra 274
The inertial subrange 277
The Lagrangian integral time scale 277
An approximate Lagrangian spectrum 278
8.6Spectra of passive scalar contaminants 279
One- and three-dimensional spectra 280
The cascade in the temperature spectrum 281
Spectra in the equilibrium range 282
The inertial-diffusive subrange 283
The viscous-convective subrange 284
The viscous-diffusive subrange 285
Summary 286
Bibliography and references 288
Index 295